On the structure of arithmetic sums of Cantor sets with constant ratios of dissection

Abstract

We investigate conditions which imply that the topological structure of the arithmetic sum of two Cantor sets with constant ratios of dissection at each step is either: a Cantor set, a finite union of closed intervals, or three mixed models (L, R and M-Cantorval). We obtain general results that apply in particular for the case of homogeneous Cantor sets, thus generalizing the results of Mendes and Oliveira. The method used here is new in this context. We also produce results regarding the arithmetic sum of two affine Cantor sets of a special kind.